Bounds on the dimension of lineal extensions

Abstract

Let $E\subseteq {\mathbb{R}}^n$ be a union of line segments and $F\subseteq {\mathbb{R}}^n$ the set obtained from $E$ by extending each line segment in $E$ to a full line. Keleti’s line segment extension conjecture posits that the Hausdorff dimension of $F$ should equal that of $E$. Working in ${\mathbb{R}}^2$, we use effective methods to prove a strong packing dimension variant of this conjecture. Furthermore, a key inequality in this proof readily entails the planar case of the generalized Kakeya conjecture for packing dimension. This is followed by several doubling estimates in higher dimensions and connections to related problems.